Empirical Methods in Finance Flashcards

Course Outline and Schedule

• The course covers various topics in financial time series analysis and econometrics.
• Characteristics of financial time series (Weeks 1 & 2): Focus on asset returns, distributions, and time dependency.
• Univariate and Multivariate time series analysis (Weeks 3 & 4): Explore time series models.
• Non-stationarity and cointegration (Week 5): Analyze non-stationary data.
• Capital Asset Pricing Model (CAPM) (Week 6): Study the Capital Asset Pricing Model.
• Multi-factor models (Week 7): Discuss multi-factor models.
• Efficient markets hypothesis (Week 8): Examine the efficient markets hypothesis.
• Volatility Modeling (Weeks 9 & 10): Explore ARCH and GARCH models.
• Non-normality and Extreme Value Theory (Weeks 11 & 12): Investigate non-normal distributions and extreme value theory.
• Multivariate GARCH models (Week 13): Discuss multivariate GARCH models.
• Dependence and copula models (Week 14): Analyze dependence using copula models.

Lecture 1: Characteristics of Financial Time Series – 1

Objectives

• Describe the variables used (Asset Returns).
• Understand main properties of asset returns:
  • Distribution (Moments).
  • Time dependency (Heteroskedasticity).
  • Correlation across assets.
• Address issues with data.

Prices vs. Returns

• Discusses the utility of using prices, log prices, and monthly log-returns of the S&P 500 index to analyze financial data.
• Statistical properties of financial returns can vary significantly based on the frequency of the data.

Prices

• Notation: P_t represents the price of a security at time t.
• Sampling Frequency:
  • Irregular: Prices observed at specific events (e.g., public offerings, dividend payments).
  • Regular: Prices observed at fixed intervals (e.g., daily, weekly).
• Frequency Types:
  • High frequency: Intraday data (tick-by-tick, 5-minute intervals).
  • Low frequency: Daily, monthly, quarterly, or annual data.
• Types of Prices:
  • Opening price: First trade of the day.
  • Closing price: Last trade of the day.
  • Lowest and Highest prices.
  • Adjusted closing price: Adjusted for corporate actions like splits and dividends.
• Typically, the end-of-period price (closing price on the last day of the period) is used.

Simple Return

• Reasons for using returns instead of prices:
  • Investors focus on returns for investment decisions.
  • Returns are easier to handle statistically.
• Formula:
  • R{t+1} = \frac{P{t+1} - Pt}{Pt}
  • Alternative form: P{t+1} = Pt(1 + R_{t+1})
  • Where:
    • P_t: Price of the asset at time t.
    • R_{t+1}: One-period simple return from date t to t+1.

Continuously Compounded (Log) Return

• If an annual interest of R{t+1}(%) is split into m payments in a year, the interest rate for each payment is R{t+1}(%)/m.
• The net value of the deposit one year later will be \left(1 + \frac{R_{t+1} \text{ (%)})}{m}\right)^m
• In the limit case where the interest is paid continuously (m \rightarrow \infty):
  • \lim{m \to \infty} \left(1 + \frac{R{t+1} \text{ (%)})}{m}\right)^m = \exp(r_{t+1})
  • With P{t+1} = Pt \times \exp(r_{t+1})
• The continuously compounded (or log) return r_{t+1} is:
  • r{t+1} = \log \left(\frac{P{t+1}}{Pt}\right) = p{t+1} - p_t
  • Where pt = \log(Pt) is the log price

Temporal Aggregation (Multiple-Period Return)

• Holding the asset for k periods from t to t+k yields the k-period simple return:
  • Rt[k] = \frac{P{t+k} - Pt}{Pt}
  • P{t+k} = Pt (1 + R_t[k])
  • = Pt \prod{j=1}^{k} (1 + R_{t+j})
  • Rt[k] = \left(\prod{j=1}^{k} (1 + R_{t+j})\right) - 1
• The k-period log return is simply the sum of continuously compounded one-period returns:
  • rt[k] = \log(1 + Rt[k]) = \log \left(\prod{j=1}^{k} (1 + R{t+j})\right) = \sum{j=1}^{k} \log(1 + R{t+j})
  • rt[k] = \sum{j=1}^{k} r_{t+j}

Contemporaneous Aggregation (Portfolio Return)

• Let p be the portfolio of N assets with weight \alpha_i on asset i.
• The simple portfolio return R_{p,t+1} is the weighted average of the simple returns of the assets:
  • R{p,t+1} = \sum{i=1}^{N} \alpha{i} R{i,t+1}
• In contrast, with continuous compounding, the log portfolio return r_{p,t+1} is not the weighted average of the log returns of the assets:
  • r{p,t+1} = \log(1 + R{p,t+1}) = \log \left(1 + \sum{i=1}^{N} \alpha{i} R{i,t+1}\right) \neq \sum{i=1}^{N} \alpha{i} r{i,t+1}
• Using simple returns or log-returns is an empirical issue, which depends in general on the problem at hand.

Dividend Payments

• For assets with periodic dividend payments, asset returns must be redefined.
  • R{t+1} = \frac{P{t+1} + D{t+1}}{Pt} - 1
  • r{t+1} = \log(P{t+1} + D{t+1}) - \log(Pt)
  • Where D_{t+1} is the dividend payment of an asset between dates t and t+1
• P{t+1} is the price of the asset at the end of t+1 (dividend not included in P{t+1}).
• Most reference indices include dividend payments but not all.
• Some reference indices (e.g., MSCI indices) are computed without (“price index”) or with dividend payments (“total return index”).
• On Datastream, stock prices are available with both definitions.

Excess Returns

• Excess return is simply the difference between the asset return and the return on the risk-free asset
  • Z{i,t+1} = R{i,t+1} - R_{0,t}
  • z{i,t+1} = r{i,t+1} - r_{0,t}
  • With R{0,t} and r{0,t} the simple return and log return of the risk-free asset.
• The index t is used because the risk-free rate between t and t+1 is determined in t
• The risk-free rate should satisfy two constraints:
  • Low risk of default (rating of the issuer)
  • Low risk of a loss of capital (choice of the maturity of the security)
• In practice, the risk-free rate is often chosen as the return of a Treasury bill with the same maturity as the investment horizon

Definition of Volatility

• The unconditional volatility is estimated as the sample standard deviation:
  • s = \sqrt{\frac{1}{T-1} \sum{t=1}^{T} (Rt - \hat{\mu})^2}
  • \sigma = \sqrt{\frac{1}{T} \sum{t=1}^{T} (Rt - \hat{\mu})^2}
  • Where R_t is the simple return on day t and \hat{\mu} is the sample mean over the T days.
• s is the unbiased estimator
• \sigma is the ML estimator, asymptotically unbiased
• For the moment, we focus on a unique estimator. We will discuss the case of a dynamic volatility later on.

Characteristics of Financial Asset Returns (Stylized Facts)

• Absence of Autocorrelations: Linear autocorrelations of asset returns are often insignificant, except for very small intraday time scales (~20 minutes, microstructure effects).
• Heavy Tails: The unconditional distribution of returns displays a power-law or Pareto-like tail, with a finite tail index greater than two.
• Gain/Loss Asymmetry: Large drawdowns are observed in stock prices, but equally large upward movements are not.
• Aggregational Normality: As the time scale increases, the distribution of returns looks more like a normal distribution. The shape varies across time scales.
• Intermittency: Returns display high variability, quantified by irregular bursts in time series of volatility estimators.
• Volatility Clustering: Volatility measures display positive autocorrelation over several days, indicating that high-volatility events tend to cluster in time.
• Conditional Heavy Tails: Even after correcting for volatility clustering (e.g., via GARCH models), residual time series still exhibit heavy tails, but less heavy than the unconditional distribution.
• Slow Decay of Autocorrelation in Absolute Returns: Autocorrelation function of absolute returns decays slowly, roughly as a power law, suggesting long-range dependence.
• Leverage Effect: Volatility measures of an asset are negatively correlated with returns of that asset.
• Volume/Volatility Correlation: Trading volume is correlated with volatility.
• Asymmetry in Time Scales: Coarse-grained measures of volatility predict fine-scale volatility better than the other way around.

Early Assumptions in Finance

• Normality of Log-Returns: Convenient for applications like the Black-Scholes model. Consistent with the Central Limit Theorem if log-returns are i.i.d.
• Time Independency (i.i.d. Process): Implied by the Efficient Market Hypothesis (EMH), which only requires unpredictability of returns.

Moments of a Random Variable

• Definitions
  • Assume X (log-returns) has the following cumulative distribution function (cdf) FX(x) = Pr[X \leq x] = \int{-\infty}^{x} f_X(u) du
  • where f_X(x) is the probability distribution function (pdf)
• Mean (Expected Value):
  • \mu = E[X] = \int{-\infty}^{\infty} x fX(x) dx
• Variance:
  • \sigma^2 = V[X] = E[(X - \mu)^2] = \int{-\infty}^{\infty} (x - \mu)^2 fX(x) dx
• k-th Non-Central Moment:
  • mk = E[X^k] = \int{-\infty}^{\infty} x^k f_X(x) dx
  • For k = 1, 2, …
  • The first non-central moment m_1 is the expected value of X.
• k-th Central Moment:
  • \muk = E[(X - m1)^k] = \int{-\infty}^{\infty} (x - m1)^k f_X(x) dx
  • For k = 1, 2, …
  • By construction, \mu_1 = 0
  • The second central moment \mu2 \equiv \sigma^2 = m2 - \mu^2 is the variance of X
  • The 3rd central moment \mu3 = E[(X - m1)^3] measures the asymmetry of the distribution
  • The 4th central moment \mu4 = E[(X - m1)^4] measures its tail-fatness.

Skewness and Kurtosis

• Standardized Skewness:
  • S[X] = E\left[\left(\frac{X - m1}{\sigma}\right)^3\right] = \frac{\mu3}{\sigma^3}
• Standardized Kurtosis:
  • K[X] = E\left[\left(\frac{X - m1}{\sigma}\right)^4\right] = \frac{\mu4}{\sigma^4}
• Interpretation:
  • Negative Skewness (S[X] < 0): Large negative realizations are more likely (crashes are more likely than booms).
  • Large Kurtosis (K[X]): Large realizations (positive or negative) are more likely.
• Normal Distribution Values:
  • Skewness = 0
  • Standardized Kurtosis = 3
  • Excess Kurtosis = K[X] - 3

Measures of location and dispersion

• Consider a log-returns \left{r_t\right}, for t = 1, …, T, which are realizations of a random variable.
• Sample Mean (Average):
  • \hat{\mu} = \bar{r} = \frac{1}{T} \sum{t=1}^{T} rt
  • The sample average is the simpler estimate of location. If the data sample is drawn from a normal distribution, the sample average is the optimal measure of location.
• Variance:
  • It is the optimal measure for normal returns.
  • s^2 = \frac{1}{T-1} \sum{t=1}^{T} (rt - \hat{\mu})^2
  • \hat{\sigma}^2 = \frac{1}{T} \sum{t=1}^{T} (rt - \hat{\mu})^2
  • s^2 is the unbiased estimator, while \hat{\sigma}^2 is the ML estimator.
• Standard Deviation:
  • The square root of the variance.
• The mean and variance are not robust to outliers.

Higher Moments

• The sample counterpart of the standardized skewness and kurtosis are
  • \hat{S} = \frac{1}{T} \sum{t=1}^{T} \left(\frac{rt - \bar{r}}{\hat{\sigma}}\right)^3
  • \hat{K} = \frac{1}{T} \sum{t=1}^{T} \left(\frac{rt - \bar{r}}{\hat{\sigma}}\right)^4
• For a normal distribution, the skewness and the excess kurtosis are zero.
• If \hat{S} < 0, the distribution has a fatter left tail.
• If \hat{S} > 0, the distribution has a fatter right tail.
• If \hat{K} < 3, the distribution has thinner tails than the normal.
• If \hat{K} > 3, the distribution has fatter tails than the normal.
• If we reject normality, we should probably use robust measures for higher moments.

Tests for Normality

• Tests:
  • Jarque-Bera test (based on the moments of the distribution).
  • Kolmogorov-Smirnov test (based on the empirical distribution function).
  • Anderson-Darling test (based on the empirical distribution function).

Jarque-Bera Test

• Based on the fact that under normality, skewness and excess kurtosis are jointly equal to zero.
• Hypotheses
  • H_0: Skewness = 0 and Kurtosis = 3
  • The normal distribution is defined by the first two moments, so there is no constraint on the mean and variance under the null.
• Under normality, the sample skewness and kurtosis are mutually independent.
• Test Statistic:
  • JB = T \left[\frac{S^2}{6} + \frac{(K - 3)^2}{24}\right]
  • Under the null hypothesis, it is asymptotically distributed as a \chi^2(2).
  • If JB \geq \chi^2_{\alpha}(2), then we reject the null hypothesis at level \alpha.

Empirical Distribution Function Goodness-of-Fit Tests

• Compare the empirical cdf and the assumed theoretical cdf F^*(x; \theta) (here, the normal distribution).
• The time series (r1, r2, …, rT) is associated with some unknown cdf, Fr(x).
• As the true distribution is unknown, it is approximated by the empirical cdf, G_T(x), defined as (step function with steps of height [1/T] at each observation)
  • GT(x) = \frac{1}{T} \sum{t=1}^{T} 1 {r_t \leq x }
• The empirical cdf G_T(x) is compared with the assumed cdf F^*(x; \theta) to see if there is good fit.
• Hypotheses
  • H0: Fr(x) = F^*(x; \theta) for all x.
  • HA: Fr(x) \neq F^*(x; \theta) for at least one value of x.
• EDF tests are based on the result that if Fr(X) is the cdf of X, then the variable Fr(X) is uniformly distributed between 0 and 1.

Kolmogorov-Smirnov and Lilliefors Tests

• A measure of the difference between two cdfs is the largest distance between the two functions G_T(x) and F^*(x; \theta).
• KS Test Statistic:
  • KS = \max{1, …, T} |F^*(x; \theta) - GT(x)|
• Recipe for the normal distribution N(\mu, \sigma^2):
  • If \mu and \sigma^2 are unknown, estimate \hat{\mu} and s^2 from the original data (Lilliefors test)
  • Sort the sample data by increasing order and denote the new sample \left{rt\right}{t=1}^{T}, with r1 \leq r2 \leq … \leq rT. By construction, we have GT(r_t) = \frac{t}{T}.
  • Evaluate the assumed theoretical cdf F^*(rt; \theta) for all values \left{rt\right}_{t=1}^{T}. Use (\mu, \sigma^2) if they are known or (\hat{\mu}, s^2) if they are not
• Compute the KS-Lilliefors test statistic:
  • KS = \max_{1, …, T} |F^*(x; \theta) - \frac{t}{T}|
  • Critical values: 0.805/\sqrt{T}, 0.886/\sqrt{T}, and 1.031/\sqrt{T} for a 10%, 5%, and 1% level test

Lecture 2: Characteristics of Financial Time Series – 2

Objectives

• Addresses the time dependency of asset returns.
• Explores the properties of correlation across asset returns.
• Discusses issues with data.

Stationarity

• Prices are generally non-stationary, while returns are typically stationary.
• Prices are often non-stationary due to economic expansion, productivity increases, or financial crises.
• Returns tend to fluctuate around a constant level, indicating a constant mean over time (mean-reverting process).
• Most return sequences can be modeled as a stochastic process with time-invariant first two moments (weak stationarity).

Serial Correlation

• No serial correlation: Autocorrelations of asset returns R_t are often insignificant, except for very small intraday time scales (≈ 20 minutes) where microstructure effects occur.
• Absence of serial correlation does not mean the returns are independent.
• Squared returns and absolute values of returns often display high serial correlation.
• The fact that returns do not show any serial correlation does not mean that they are independent!
  • In fact, squared returns and the absolute value of returns display high serial correlation

Time Dependence of Moments

• The time dependence of moments is crucial for testing market efficiency.
• Assuming a weakly (or covariance) stationary time series (rt, …, rT), we have:
  • E[r_t] = \mu
  • \forall t V[r_t] = \sigma^2
  • \forall t Cov[rt, r{t-k}] = \gamma_k
  • \forall t, k
• \gammak is the auto-covariance, measuring the dependency of rt with respect to its past observation r_{t-k}.
  • \gamma0 = V[rt] = E[(r_t - \mu)^2]
  • \gammak = \gamma{-k}
• For a weakly stationary process, the autocovariance \gamma_k depends on the horizon k, but not on the date t.

Auto-Correlation Function (ACF)

• The auto-correlation function (ACF) is preferred over auto-covariance as a measure of dependency because it is a bounded measure:
  • \rhok = \frac{Cov[rt, r{t-k}]}{\sqrt{V[rt]V[r{t-k}]}} = \frac{Cov[rt, r{t-k}]}{V[rt]} = \frac{\gammak}{\gamma0}
  • \rhok = \rho{-k}, \rho0 = 1, -1 \leq \rhok \leq 1
• A consistent estimator of \rho_k is:
  • \hat{\rho}k = \frac{\frac{1}{T-k} \sum{t=k+1}^{T} (rt - \bar{r})(r{t-k} - \bar{r})}{\frac{1}{T-1} \sum{t=1}^{T} (rt - \bar{r})^2}
• Asymptotic properties of \hat{\rho}k are complex due to its nature as a ratio of quadratic functions of rt.

Asymptotic Distribution of Serial Correlation

• General Case:
  • The asymptotic distribution of (\hat{\rho}1, …, \hat{\rho}K) is: \sqrt{T} \begin{bmatrix} \hat{\rho}1 - \rho1 \ \vdots \ \hat{\rho}K - \rhoK \end{bmatrix} \sim N \left( \begin{bmatrix} 0 \ \vdots \ 0 \end{bmatrix}, V \right)
  • V{i,j} = \sum{s=-\infty}^{\infty} (\rho{s+i} + \rho{s-i} - 2\rhoi \rhos)(\rho{s+j} + \rho{s-j} - 2\rhoj \rhos)
• Case 1: r_t is an i.i.d. process:
  • V[\hat{\rho}_k] = \frac{1}{T}
  • \sqrt{T} \hat{\rho}_k is asymptotically normal N(0,1) for all k > 0.
• Case 2: r_t is a stationary moving average MA(q) process:
  • rt = \mu + \sum{i=1}^{q} \thetai \epsilon{t-i}, with \theta0 = 1 and \rhok = 0 for k > q.
  • V[\hat{\rho}k] = \frac{1}{T} \left[1 + 2 \sum{i=1}^{q} \rho_i^2 \right]
  • \sqrt{T} \hat{\rho}k is asymptotically normal N(0,V) with V = 1 + 2 \sum{i=1}^{q} \rho_i^2 for all k > q.
  • We test \hat{\rho}_k = 0 by assuming that the process is an MA(k-1) to estimate V.

Box-Pierce / Ljung-Box Tests

• Hypothesis:
  • Null hypothesis: H0 : \rho1 = … = \rho_p = 0
  • Alternative hypothesis: H1 : \rhok \neq 0 for some k = 1,…, p.
• Box-Pierce Q-Statistic:
  • Qp = T \sum{i=1}^{p} \hat{\rho}_i^2
  • Under the null hypothesis H0 : \rho1 = … = \rhop = 0, all the \hat{\rho}k have V[\hat{\rho}_k] = \frac{1}{T}.
  • Therefore, Q_p is asymptotically distributed as \chi^2(p).
  • If Qp \geq \chi^2{\alpha}(p), then we reject the null hypothesis at level \alpha.
• Ljung-Box Q’ Statistic:
  • Q'p = T(T + 2) \sum{j=1}^{p} \frac{1}{T - j} \hat{\rho}_j^2
  • Has better finite-sample properties, with the same asymptotic distribution.

Volatility Clustering

• Volatility clustering implies that large price changes (returns with large absolute values or squared returns) occur in clusters.
• Large price changes tend to be followed by large price changes.
• Periods of tranquility alternate with periods of high volatility.
• Volatility clustering is a consequence of the autocorrelation of the squared returns.

Serial Correlation in Volatility

• To test for time dependency in volatility, we need a time-varying measure of volatility.
• Two possible measures:
  • Mean-adjusted squared returns.
  • Absolute returns.
• Assume returns have the following dynamics:
  • rt = \mu + \epsilont with \epsilont = \sigmat z_t
    • \mu is the constant mean.
    • \epsilon_t the unexpected returns.
    • \sigma_t the time-varying volatility based on information at date t–1.
    • z_t is an N(0,1) innovation.

Proxies for Conditional Volatility

1. Conditional Expectation of Squared Returns:

• Given information set I_{t-1}.
  • E[\epsilont^2 | I{t-1}] = E[\sigmat^2 zt^2 | I{t-1}] = \sigmat^2 E[zt^2 | I{t-1}] = \sigma_t^2
  • Because z_t^2 is distributed as a \chi^2(1).
  • \epsilon_t^2 can be viewed as a proxy for volatility at time t.

2. Conditional Expectation of Absolute Returns:

• If \epsilont \sim N(0, \sigmat^2), then:
  • E[|\epsilont| | I{t-1}] = \sigma_t \sqrt{2/\pi}.
  • |\epsilont| / \sqrt{2/\pi} can be used as a proxy for \sigmat.
• Both measures(1 and 2) are noisy estimators of conditional volatility.

Volatility Asymmetry

• Volatility is more affected by negative news than positive news.
• Captured through regressions:
  • \epsilont^2 = \omega + \beta \epsilon{t-1}^2 + \gamma \epsilon{t-1}^2 \times 1{{\epsilon_{t-1} < 0}} or
  • \epsilont = \omega + \beta \epsilon{t-1} + \gamma \epsilon{t-1} \times 1{{\epsilon_{t-1} < 0}}
    • \beta: Direct effect of past returns.
    • \gamma: Additional impact of negative return shocks.

Cross-Correlation

• Covariance between two series r{i,t} and r{j,t} is defined as:
  • Cov[r{i,t}, r{j,t}] = \frac{1}{T} \sum{t=1}^{T} (r{i,t} - \mui)(r{j,t} - \mu_j)
    • \mui and \muj are the sample means of r{i,t} and r{j,t} .
• Correlation between r{i,t} and r{j,t} is defined as:
  • \rho[r{i,t}, r{j,t}] = \frac{Cov[r{i,t}, r{j,t}]}{\sqrt{V[r{i,t}]V[r{j,t}]}}
  • By construction, \rho[r{i,t}, r{i,t}] = 1, \rho[r{i,t}, r{j,t}] = \rho[r{j,t}, r{i,t}] and -1 \leq \rho[r{i,t}, r{j,t}] \leq 1
• The correlation is a measure of linear association. Series may be uncorrelated but not independent.

Estimation of Cross-Correlation

• The correlation between r{i,t} and r{j,t} is estimated as:
  • \hat{\rho}[r{i,t}, r{j,t}] = \frac{\sum{t=1}^{T} (r{i,t} - \hat{\mu}i)(r{j,t} - \hat{\mu}j)}{\sqrt{\sum{t=1}^{T} (r{i,t} - \hat{\mu}i)^2 \sum{t=1}^{T} (r{j,t} - \hat{\mu}_j)^2}}

Rejection of Previous Assumptions

• For actual returns, the assumptions of normality and time independency have been widely rejected.
  • Asymmetry.
  • Fat tails.
  • Aggregated normality.
  • No serial correlation.
  • Volatility clustering.
  • Asymmetric volatility.
  • Time-varying correlation.

Portfolio

• N risky securities
• Securities returns: 𝑅#)! = (𝑅!,#)!, … , 𝑅6,#)!)′
• Expected returns: 𝜇#)! = 𝐸[𝑅#)!] = (𝜇!,#)!, … , 𝜇6,#)!)′ (measured in t)
• Covariance matrix: Σ#)! = 𝐸[(𝑅#)! − 𝜇#)!)(𝑅#)! − 𝜇#)!)5] = I 𝜎!,#)! $ … 𝜎!6,#)! ⋮ ⋱ ⋮ 𝜎!6,#)! … 𝜎6,#)! $ J
• Portfolio weights: 𝛼# = (𝛼!,#, … , 𝛼6,#)′, with 𝛼# 5𝑒 = ∑ 𝛼+,# 6 +(! = 1, 𝑒 = (1, … ,1)′
• Ex-post portfolio return: 𝑅2,#)! = 𝛼# 5𝑅#)! = ∑ 𝛼+,#𝑅+,#)! 6 +(! (Realized return)
• Ex-ante portfolio expected return: 𝐸b𝑅2,#)!c = 𝜇2,#)! = 𝛼# 5𝜇#)! = ∑ 𝛼+,#𝜇+,#)! 6 +(!
• Ex-ante portfolio variance: 𝑉b𝑅2,#)!c = 𝜎2,#)! $ = 𝛼# 5Σ#)!𝛼# = ∑ ∑ 𝛼+,#𝛼-,#𝜎+